scholarly journals A New Second-Order Iteration Method for Solving Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Shin Min Kang ◽  
Arif Rafiq ◽  
Young Chel Kwun
Keyword(s):  
Nonlinear Equations ◽  
Iteration Method ◽  
Nonlinear Operator ◽  
Efficiency Index ◽  
Second Order ◽  
First Order ◽  
Newton Raphson ◽  
Second Order Convergence ◽  
Derivatives Of ◽  
Raphson Method

We establish a new second-order iteration method for solving nonlinear equations. The efficiency index of the method is 1.4142 which is the same as the Newton-Raphson method. By using some examples, the efficiency of the method is also discussed. It is worth to note that (i) our method is performing very well in comparison to the fixed point method and the method discussed in Babolian and Biazar (2002) and (ii) our method is so simple to apply in comparison to the method discussed in Babolian and Biazar (2002) and involves only first-order derivative but showing second-order convergence and this is not the case in Babolian and Biazar (2002), where the method requires the computations of higher-order derivatives of the nonlinear operator involved in the functional equation.

scholarly journals An Improved Secant Algorithm of Variable Order to Solve Nonlinear Equations Based on the Disassociation of Numerical Approximations and Iterative Progression

2020 ◽  
Vol 12 (6) ◽  
pp. 50
Author(s):  
Christian Vanhille
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Secant Method ◽  
Variable Order ◽  
Discretization Step ◽  
Newton Raphson ◽  
Analytic Expression ◽  
The One ◽  
Derivatives Of ◽  
Raphson Method

We propose an iterative method to evaluate the roots of nonlinear equations. This Secant-based technique approximates the derivatives of the function numerically through a constant discretization step h disassociated from the iterative progression. The algorithm is developed, implemented, and tested. Its order of convergence is found to be h-dependent. The results obtained corroborate the theoretical deductions and evidence its excellent behavior. For infinitesimal h-values, the algorithm accelerates the convergence of the Secant method to order 2 (the one of the Newton-Raphson method) with no need for analytic expression of derivatives (the advantage of the Secant method).

scholarly journals PERBANDINGAN SOLUSI SISTEM PERSAMAAN NONLINEAR MENGGUNAKAN METODE NEWTON-RAPHSON DAN METODE JACOBIAN

2013 ◽  
Vol 2 (2) ◽  
pp. 11
Author(s):  
NANDA NINGTYAS RAMADHANI UTAMI ◽  
I NYOMAN WIDANA ◽  
NI MADE ASIH
Keyword(s):  
Nonlinear Equations ◽  
Iteration Method ◽  
Initial Point ◽  
Iteration Process ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Newton Raphson ◽  
Raphson Method

System of nonlinear equations is a collection of some nonlinear equations. The Newton-Raphson method and Jacobian method are methods used for solving systems of nonlinear equations. The Newton-Raphson methods uses first and second derivatives and indeed does perform better than the steepest descent method if the initial point is close to the minimizer. Jacobian method is a method of resolving equations through iteration process using simultaneous equations. If the Newton-Raphson methods and Jacobian methods are compared with the exact value, the Jacobian method is the closest to exact value but has more iterations. In this study the Newton-Raphson method gets the results faster than the Jacobian method (Newton-Raphson iteration method is 5 and 58 in the Jacobian iteration method). In this case, the Jacobian method gets results closer to the exact value.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

scholarly journals The Flow Separation of Peristaltic Transport for Maxwell Fluid between Two Coaxial Tubes

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
S. Z. A. Husseny ◽  
Y. Abd elmaboud ◽  
Kh. S. Mekheimer
Keyword(s):  
Shear Stress ◽  
Flow Separation ◽  
Maxwell Fluid ◽  
Maxwell Model ◽  
Point Of View ◽  
Nonlinear Pde ◽  
First Order ◽  
Newton Raphson ◽  
Catheter Tube ◽  
Raphson Method

We study the peristaltic mechanism of an incompressible non-Newtonian biofluid (namely, Maxwell model) in the annular region between two coaxial tubes. The inner tube represents the endoscope tube. The system of the governing nonlinear PDE is solved by using the perturbation method to the first order in dimensionless wavenumber. The modified Newton-Raphson method is used to predict the flow separation points along the peristaltic wall and the endoscope tube. The results show that the presence of the endoscope (catheter) tube in the artery increases the pressure gradient and shear stress. Such a result seems too reasonable from the physical and medical point of view.

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